Encyclopedia of Microtonal Music Theory

combination product set (CPS)

[John Chalmers, Divisions of the Tetrachord]

Harmonically symmetrical musical structures invented by Ervin M. Wilson in the late 1960's. CPS's are generated by taking the products of n harmonic factors m at a time and reducing the derived tones to a common octave.

The best studied sets are the six tone Hexany (n=4, m=2) and the 20 element Eikosany (n=6, m=3).

CPS may be partitioned into sets of inversionally related chords, triads in the case of the Hexany and tetrads in the Eikosany. CPS are also characterized by intervallically symmetrical melodic properties and may be defined in equal temperaments as well as Just Intonation.

Erv Wilson showed how Partch's scales could be geometrically depicted, with the tonic lying in the center of a fantastically symmetrical shape. Erv's constructions relied on a concept called the lattice: every pitch in a JI tuning system can be mapped to a point in space (sometimes higher-dimensional space), such that its nearest neigbors are the pitches most consonant with it.

Erv Wilson also created new scales which were the perfect theoretical complement to Partch's Tonality Diamonds. Rather than containing one central pitch, the new scales put every note in the scale in an equal position, each sharing an equal number of consonances with its neighbors. Yet there is no "space" in the lattice into which any central (or more central) pitch could be inserted. He called these scales "CPS scales". Very mathematical.